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Chapter 14 Fluids
Mass Density
Pressure
Pressure in a Static Fluid
Pascal's Principle
Archimedes' Principle
Fluids in Motion
The Equation of Continuity
DEFINITION OF MASS DENSITY
The mass density ρ is the mass m of a substance divided by its volume V:
(11.1)
SI Unit of Mass Density: kg/m3
Table 11.1 Mass Densitiesa of Common Substances
Mass Density
Substance
ρ (kg/m3)
Solids
Aluminum
2 700
Brass
8 470
Concrete
2 200
Copper
8 890
Diamond
3 520
Gold
19 300
Ice
917
Iron (steel)
7 860
Liquids
Blood (whole, 37 °C)
1 060
Ethyl alcohol
806
Mercury
13 600
Oil (hydraulic)
800
Water (4 °C)
1.000 × 103
Gases
Air
1.29
Carbon dioxide
1.98
Helium
0.179
Hydrogen
0.0899
Nitrogen
1.25
Oxygen
1.43
a
Unless otherwise noted, densities are given at 0 °C and 1 atm pressure.
Pressure
Water applies a force
perpendicular to each
surface within the water,
including the walls and
bottom of the swimming
pool, and all parts of the
swimmer’s body.
The pressure P exerted by a fluid is defined as the magnitude F of the force acting
perpendicular to a surface divided by the area A over which the force acts:
(11.3)
The SI unit for pressure is a newton/meter2 (N/m2), a combination that is referred to as a
pascal (Pa).
Pressure in a Static Fluid
Since the column is in equilibrium, the sum of the vertical forces equal to zero:
The mass m is related to the density ρ and the volume V of the column by m = ρV= ρAh.
The pressure difference between two points in the fluid only depends on the
vertical height between the two points, not the horizontal distance within the
fluid
Example 4 The Swimming Hole
Figure 11.8 shows the cross section of a swimming hole. Points A and B are
both located at a distance of h = 5.50 m below the surface of the water. Find the
pressure at each of these two points.
PASCAL’S PRINCIPLE
Any change in the pressure applied to a completely enclosed fluid is transmitted
undiminished to all parts of the fluid and the enclosing walls.
Buoyant force
ARCHIMEDES’ PRINCIPLE
Any fluid applies a buoyant force to an
object that is partially or completely
immersed in it; the magnitude of the buoyant
force equals the weight of the fluid that the
object displaces:
The liquid applies to the cylinder a net
upward force, or buoyant force, whose
magnitude FB is
Example 10 A Goodyear Airship
Normally, a Goodyear airship, such as that in Figure, contains about 5.40 × 103
m3 of helium (He) whose density is 0.179 kg/m3. Find the weight of the load WL
that the airship can carry in equilibrium at an altitude where the density of air is
1.20 kg/m3.
Check Your Understanding 2
A glass is filled to the brim with water and has an ice cube floating in it. When
the ice cube melts, what happens?
a. Water spills out of the glass.
b. The water level in the glass drops.
c. The water level in the glass does not change.
Example 18 The Buoyant Force
A father (weight W = 830 N) and his daughter (weight W = 340 N) are spending
the day at the lake. They are each sitting on a beach ball that is just submerged
beneath the water (see Figure). Ignoring the weight of the air within the balls
and the parts of their legs that are underwater, find the radius of each ball.
Fluids in Motion
• The steady flow the velocity of the fluid particles at any point is constant as
time passes. The steady flow is often called streamline flow.
• The incompressible fluid is that the density of a liquid remains almost
constant as the pressure changes
• A nonviscous fluid flows in a manner with no dissipation of energy.
Figure 11.25 At
any point along a
streamline, the velocity
vector of the fluid
particle at that point is
tangent to the
streamline.
The mass flow rate: is the mass of fluid per second that flows through a tube.
EQUATION OF CONTINUITY
The mass flow rate (ρAv) has the same value at every position along a tube that has a single entry
and a single exit point for fluid flow. For two positions along such a tube
(11.8)
where
ρ = fluid density (kg/m3)
A = cross-sectional area of tube (m2)
v = fluid speed (m2)
SI Unit of Mass Flow Rate: kg/s
For an incompressible fluid, ρ1 = ρ2,
Example 11 A Garden Hose
A garden hose has an unobstructed opening with a cross-sectional area of 2.85 ×
10–4 m2, from which water fills a bucket in 30.0 s. The volume of the bucket is
8.00 × 10–3 m3 (about two gallons). Find the speed of the water that leaves the
hose through (a) the unobstructed opening and (b) an obstructed opening with
half as much area.
Check Your Understanding 3
Water flows from left to right through the five sections (A, B, C, D, E) of the
pipe shown in the drawing. In which section(s) does the water speed increase,
decrease, and remain constant? Treat the water as an incompressible fluid.